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In[383]:= (*QUESTION 6*)
(*Mean*)
f = 1 2 Sin[x];
xbar = Integrate[f * x, {x, 0, Pi}] // N (* Mean *)
(* Variance *)
var = Integratef * x - xbar^2, {x, 0, Pi} // N
(*Skewness*)
s = FullSimplifyIntegratef * x - xbar^3, {x, 0, Pi} // N
(*Kurtosis*)
k = FullSimplifyIntegratef * x - xbar^4, {x, 0, Pi} // N
sigma = sqrt[var] // N;
(*for same mean and standard dev, skewness and kurtosis for the Normal Distribution*)
(*Skewness for Normal Distribution*)
S = Integratex - xbar^31
sigma * 2 PiExp
-x - xbar^2
2 * sigma^2, {x, 0, Pi} // N
(*Kurtosis for Normal Distribution*)
K = Integratex - xbar^(4)1
sigma * 2 PiExp
-x - xbar^2
2 * sigma^2, {x, 0, Pi} // N
Out[384]= 1.5708
Out[385]= 0.467401
Out[386]= 0.
Out[387]= 0.479255
Out[389]= 0.
Out[390]= 3. Erf 1.11072sqrt[0.467401]
sqrt[0.467401]4 +
2.71828- 1.2337
sqrt0.4674012 -3.09243 sqrt[0.467401] - 3.75994 sqrt[0.467401]3
(*While the skewness for both the curves is the same, zero,
the kurtosis for the Normal distribution is greater than that for the given function.*)
(*QUESTION 7*)
(*For N = 2 *)
list1 = {}; sum = 0;
For[i = 1, i < 500, i++,
For[j = 1, j < 2, j++,
sum = sum + RandomVariate[UniformDistribution[{0, 1}]]
]
AppendTo[list1, sum]; sum = 0;
]
Histogram[list1, 20]
(*For N = 8 *)
list2 = {}; sum = 0;
For[i = 1, i < 500, i++,
For[j = 1, j < 8, j++,
sum = sum + RandomVariate[UniformDistribution[{0, 1}]]
]
AppendTo[list2, sum]; sum = 0;
]
Histogram[list2, 20]
(*For N = 20 *)
list3 = {}; sum = 0;
For[i = 1, i < 500, i++,
For[j = 1, j < 20, j++,
sum = sum + RandomVariate[UniformDistribution[{0, 1}]]
]
AppendTo[list3, sum]; sum = 0;
]
Histogram[list3, 20]
(*For N = 50 *)
list4 = {}; sum = 0;
For[i = 1, i < 500, i++,
For[j = 1, j < 50, j++,
sum = sum + RandomVariate[UniformDistribution[{0, 1}]]
]
AppendTo[list4, sum]; sum = 0;
]
Histogram[list4, 20]
2 Assignment_Solution.nb
Out[365]=
0.2 0.4 0.6 0.8 1.00
5
10
15
20
25
30
Out[368]=
2 3 4 50
10
20
30
40
50
Out[371]=
6 7 8 9 10 11 12 130
10
20
30
40
50
60
70
Assignment_Solution.nb 3
Out[374]=
20 22 24 26 28 300
10
20
30
40
50
4 Assignment_Solution.nb
Assignment_Solution.nb 5